Functional Limit Theorems for Hawkes Processes

TL;DR

This paper establishes functional limit theorems for Hawkes processes, revealing how their long-term behavior depends on the dispersion of child events and differentiating between subcritical, critical, and heavily dispersed cases.

Contribution

It provides new limit theorems for Hawkes processes under minimal conditions and characterizes their behavior across different regimes of dispersion.

Findings

Subcritical Hawkes processes follow FLLNs and FCLTs depending on dispersion.

Critical processes with weak dispersion behave like CIR processes without mean reversion.

Heavily dispersed critical processes exhibit long-range dependencies similar to subcritical ones.

Abstract

We prove that the long-run behavior of Hawkes processes is fully determined by the average number and the dispersion of child events. For subcritical processes we provide FLLNs and FCLTs under minimal conditions on the kernel of the process with the precise form of the limit theorems depending strongly on the dispersion of child events. For a critical Hawkes process with weakly dispersed child events, functional central limit theorems do not hold. Instead, we prove that the rescaled intensity processes and rescaled Hawkes processes behave like CIR-processes without mean-reversion, respectively integrated CIR-processes. We provide the rate of convergence by establishing an upper bound on the Wasserstein distance between the distributions of rescaled Hawkes process and the corresponding limit process. By contrast, critical Hawkes process with heavily dispersed child events share many properties of subcritical ones. In particular, functional limit theorems hold. However, unlike subcritical processes critical ones with heavily dispersed child events display long-range dependencies.